Method to transiently detect sample features using cantilevers

ABSTRACT

An approach to determine cantilever movement is presented. An observer based state estimation and statistical signal detection and estimation techniques are applied to Atomic Force Microscopes. A first mode approximation model of the cantilever is considered and an observer is designed to estimate the dynamic states. The cantilever-sample interaction is modeled as an impulsive force applied to the cantilever in order to detect the presence of sample. A generalized likelihood ratio test is performed to obtain the decision rule and the maximum likelihood estimation of the unknown arrival time of the sample profile and unknown magnitude of it. The use of the transient data results in sample detection at least ten times faster than using the steady state characteristics.

CROSS-REFERENCE TO RELATED PATENT APPLICATIONS

This patent application is a continuation-in-part of U.S. patentapplication Ser. No. 10/953,195, filed Sep. 29, 2004, now U.S. Pat. No.7,066,014 which claims the benefit of U.S. Provisional PatentApplication No. 60/507,409, filed Sep. 30, 2003.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT

This invention was made in part with Government support under GrantNumber 9733802 awarded by the National Science Foundation. TheGovernment has certain rights in this invention.

FIELD OF THE INVENTION

This invention pertains to surface image measurements and moreparticularly relates to imaging/detection of sample features usingcantilevers.

BACKGROUND OF THE INVENTION

Imaging and detection of sample features using cantilevers have been inuse for decades. For example, one prevalent use of cantilevers is inScanning Probe Microscopes (SPM's), which are instruments that have beenin use in universities and industrial research laboratories since theearly 1980's. These instruments allow for various imaging of surfaces aswell as measurement of the intermolecular forces between two surfaces(or a small tip and a flat surface) in vapors or liquids with a distanceresolution of 1 Å. This means that images and forces can be obtained atthe atomic level. Over the years, the technique has been improved andits scope extended so that it is now capable of measuring many differentsurface properties and phenomena. One type of SPM is an atomic forcemicroscope (AFM), which generally consists of a sample holder and aprobe that is supported at the end of a force-measuring cantileverspring. The AFM measures a local property such as height, opticalabsorption, or magnetism, with a probe or “tip” placed very close to asample. It operates by first positioning the tip near the surface andthen moving the tip or surface vertically to maintain a constantdeflection or amplitude of the cantilever in contact and tapping mode)respectively while measuring the force produced on the tip by thesurface. The sample is moved in a lateral direction to perform a rasterscan of the sample surface. The force is calculated by measuring thedeflection of the cantilever spring supporting the tip.

The most common method of measuring deflection of the cantilever is theoptical beam deflection method where vertical deflection can generallybe measured with picometer resolution. The method works by reflecting alaser beam off end of the cantilever. Angular deflection of thecantilever causes a twofold larger angular deflection of the laser beam.The reflected beam strikes a split photodiode (i.e., two side-by-sidephotodiodes) and the difference between the two photodiode signalsindicates the position of the laser beam on the split photodiode andthus the angular deflection of the cantilever.

FIGS. 23 a and b illustrate a typical setup of a cantilever in an AFMapplication. The laser 1000 outputs a laser beam 1002 that is pointed atthe cantilever 1004. A piezoelectric scanner 1006 is used to positionthe sample. The laser beam 1002 deflects off the cantilever 1004 and isreflected into the split photo-diode 1008 via mirror 1010. The output ofthe split photo-diode is conditioned via module 1012 and is input intofeedback control module 1014 that is used to control the position of thesample movement of piezoelectric scanner 1006. In static forcespectroscopy the cantilever deflection is solely due to the tip-sampleinteraction. The piezoelectric scanner 1006 is rastered in the lateraldirections and the deflection of the tip is used to interpret sampleproperties. In the dynamic mode, the cantilever support 1016 is forcedsinusoidally using a dither piezo 1018. The changes in the oscillationscaused by the sample are interpreted to obtain its properties.

The cantilever has low stiffness and high resonant frequency that allowsit to probe interatomic forces. Micro-cantilevers, which are cantilevershaving lengths typically ranging from 100 to 200 μm with tips of 5 nm,have been utilized in biological sciences to perform feats such ascutting DNA strands and monitoring RNA activity. Another application ofthe micro-cantilever is in the detection of single electron spin thathas significant ramifications for quantum computing technology.

In spite of the underlying promise, considerable challenges remain.Pivotal to harnessing the vast potential of micro-cantilever basedtechnology is ultra-fast interrogation capabilities. This is apparent asthe manipulation, interrogation and control of atoms or spins ofelectrons needs to be accomplished for material that has macroscopicdimensions. To achieve high throughput, fast interrogation isimperative. It is becoming increasingly evident that for manynanotechnological studies, high bandwidth is a necessity. For example,in the field of cell biology, proposals on using nanotechnology havebeen presented where nano-probes track events in the cell. These eventsoften have time-scales in the micro-second or nano-second regimes.Current measurement techniques do not meet the aforementioned highprecision and bandwidth requirements. For example, the micro-cantileveris often operated in the dynamic mode where it is forced sinusoidally bya dither-piezo. This mode of operation has advantages of better signalto noise ratio and being gentle on the sample. Most dynamic imagingmethods employing micro-cantilevers currently use variables such as theamplitude and phase, or the equivalent frequency of the micro-cantileverto infer sample characteristics. These are steady state characteristicsand do not hold much significance during the transient of the cantileveroscillation. The present methods are therefore inherently slow owing tothe large settling times of the cantilever oscillations.

BRIEF SUMMARY OF THE INVENTION

The invention provides an ultra-fast method to obtain imaging data usingtransient data of a cantilever used in various applications such as anatomic force microscope (AFM), computing technologies, etc. Thecantilever is modeled via an observer that is used to estimate the stateof the model. Corresponding data during the transient state of thecantilever is analyzed using a likelihood ratio test to detect and probethe sample profile. The use of the transient data results in sampledetection at least ten times faster than using the steady statecharacteristics. The time taken to detect the sample is proportional tothe natural frequency of the cantilever and effectively decoupled fromthe quality factor of the cantilever.

These and other advantages of the invention, as well as additionalinventive features, will become more apparent from the followingdetailed description when taken in conjunction with the accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram generally illustrating an exemplary atomicforce microscope environment on which the present invention may reside;

FIG. 2 is a block diagram generally illustrating the control blocks ofthe present invention;

FIG. 3 is a block diagram of a spring-damper-mass model of cantileverdynamics;

FIG. 4 is a plot illustrating the effect of cantilever-sampleinteraction on the cantilever;

FIG. 5 is a block diagram of the observer based detection method inaccordance with the teachings of the present invention;

FIG. 6 is a block diagram of a cantilever-sample interaction model inaccordance with the teachings of the present invention;

FIG. 7 is diagram illustrating the tracking of the observer of FIG. 5with respect to the cantilever deflection signal;

FIG. 8 is a graph of the deflection signal of a cantilever when thecantilever hits a sample;

FIG. 9 is a plot of the error in estimation of the observer of FIG. 5and a demodulated cantilever deflection signal of FIG. 8;

FIG. 10 is a plot showing the tip-sample interaction force of thecantilever of FIG. 1 when the cantilever hits a sample of 4 nm height;

FIG. 11 is a plot showing innovations from the observer of FIG. 5 and anestimated dynamic profile in accordance with the teachings of thepresent invention;

FIG. 12 is a plot showing innovations from the observer of FIG. 5 inaccordance with the teachings of the present invention when thecantilever of FIG. 1 has a single hit with a sample of 2 nm and 4 nmheight;

FIG. 13 is a plot showing innovations from the observer of FIG. 5 inaccordance with the teachings of the present invention when thecantilever of FIG. 1 has multiple hits with a sample of 2 nm and 4 nmheight;

FIG. 14 is a plot showing innovations from the observer of FIG. 5 whenthe cantilever of FIG. 1 has a single hit with a sample of 4 nm heightand a corresnow ponding generalized likelihood ratio and decision rulein accordance with the teachings of the instant invention;

FIG. 15 is a plot showing innovations from the observer of FIG. 5 whenthe cantilever of FIG. 1 has multiple hits with a sample of 4 nm heightand a corresponding generalized likelihood ratio and decision rule inaccordance with the teachings of the instant invention;

FIG. 16 is a plot showing innovation sequence data from the observer ofFIG. 5 and an estimated dynamic profile in accordance with the teachingsof the instant invention with a sample of 2 nm height;

FIG. 17 is a plot showing innovation sequence data from the observer ofFIG. 5 and an estimated dynamic profile in accordance with the teachingsof the instant invention with a sample of 4 nm height;

FIGS. 18 a and 18 b are graphs illustrating the model response and thepiezo response of an atomic force microscope in accordance with theinvention;

FIGS. 19 a and b are graphs illustrating an input voltage and the piezoresponse to the input voltage;

FIGS. 20 a and 20 b are graphs illustrating the cantilever position andinnovation process in response to a pulse shape generated using thepiezo dynamics;

FIGS. 21 a and 21 b are graphs illustrating the innovation response fora piezo oscillation;

FIGS. 22 a-c are graphs illustrating the innovation and likelihood ratiowith respect to a piezo movement;

FIG. 23 a is a block diagram of a prior art atomic force microscope; and

FIG. 23 b is a diagram of the cantilever of FIG. 18 a in contact with asample.

DETAILED DESCRIPTION OF THE INVENTION

The invention provides a method to utilize the transient behavior of thecantilever oscillations for sample imaging and detection in sensingsystems (i.e., systems that detect and/or image a sample). The transientbased methods of the invention improve performance of interrogationspeeds by approximately two orders of magnitude over steady state basedmethods. Fundamental limitations due to high quality factors are removedby estimating cantilever motion. A model of the cantilever is used tobuild an observer that is used to estimate the state of the cantileverdynamics. The cantilever-sample interaction is modeled and correspondingdata during the transient state of the cantilever is analyzed usinggeneralized likelihood ratio test to detect and probe the sampleprofile. The use of the transient data results in sample detection atleast two orders faster than using the steady state data based methods.

Turning to the drawings, wherein like reference numerals refer to likeelements, the invention is illustrated as being implemented in asuitable sensing environment 100. In the environment, a laser 102outputs a laser beam 104 that is pointed at the cantilever 106. Thelaser beam 104 deflects off the cantilever 106 and is reflected into asplit photo-diode 110 via mirror 112. The output of the splitphoto-diode is conditioned via module 114 and is input into a controlmodule 116 that is used to control the position of the sample relativeto the cantilever 106. In one embodiment, the relative position of thesample is controlled by a positioning device 108. Alternatively, thecantilever position may be controlled in relation to the sample. In thedescription that follows, an AFM shall be used to describe theinvention. While an AFM shall be used, the invention may be used inother sensing applications where a cantilever is used to detect and/orscan a surface of a sample. For example, the techniques and methodsdescribed herein may be used in detecting features in systems, detectingstructural problems in structures, detecting events such as when asample is changing, etc.

In an AFM, a piezoelectric scanner is used to position the sample in oneembodiment. The laser beam 104 deflects off the cantilever 106 and isreflected into a split photo-diode 110 via mirror 112. The output of thesplit photo-diode is conditioned via module 114 and is input intocontrol module 116 that is used to control the position of the sample bymovement of piezoelectric scanner 108. The piezoelectric scanner 108 israstered in the lateral directions and the deflection of the cantileveris used to interpret sample properties. In the dynamic mode, the ditherpiezo 118 is used to force the cantilever support 120 to movesinusoidally. The changes in the oscillations caused by the sample areinterpreted to obtain its properties.

Turning now to FIG. 2, a block diagram of the system model that thecontrol module 116 implements is shown. The cantilever module 200 is themodel of the cantilever. The cantilever module 200 has inputs g(t) andu(t) where g(t) is the external forcing on the cantilever and u(t) isthe thermal noise. The model of the cantilever when it is forcedsinusoidally at its first resonant is accurately described by{umlaut over (p)}+2ξω₀ {dot over (p)}+ω ₀ ² p=g(t)+u(t)  (1)where p(t), g(t), u(t) ω₀ and ξ denote the micro-cantilever deflectionas measured by the photo-diode 110, the external forcing on thecantilever, forcing due to thermal noise, the first resonant frequencyof the cantilever and the damping factor in free medium respectively.This model can be viewed as a spring mass damper system as illustratedin FIG. 3. Turning briefly to FIG. 3, the spring-damper-mass modeldepicts the cantilever dynamics oscillating freely (Φ=0) at itsresonance frequency as described by Equation (1) with

$\begin{matrix}{{{\omega_{0} = \sqrt{\frac{k}{m}}},{{2{\xi\omega}_{0}} = \frac{c}{m}},{and}}{{g(t)} = {\frac{k}{m}{b(t)}}}} & (2)\end{matrix}$where k is the spring constant, c is the damping constant, m is themass, and b(t) is the sinusoidal forcing due to the dither piezo.

Note that

$\xi = \frac{1}{2Q}$with Q being the quality factor of the micro-cantilever. The parametersof the cantilever model can be accurately obtained by thermal noiseanalysis. From Equation 1, the continuous time state-space model of themicro-cantilever dynamics can be described as,

$\begin{matrix}{{\begin{pmatrix}{\overset{.}{x}}_{1} \\{\overset{.}{x}}_{2}\end{pmatrix} = {{\begin{pmatrix}0 & 1 \\{- \omega_{0}^{2}} & {{- 2}{\xi\omega}_{0}}\end{pmatrix}\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix}} + {\begin{pmatrix}0 \\1\end{pmatrix}\left( {u + f_{s}} \right)}}}{y = {{\begin{pmatrix}1 & 0\end{pmatrix}\begin{pmatrix}x_{1} \\x_{2}\end{pmatrix}} + \upsilon}}} & (3)\end{matrix}$where state x₁ denotes the cantilever position p(t), state x₂ denotesthe cantilever velocity {dot over (p)}(t), and υ denotes the noiseaffecting the photodiode sensor. u=n_(th)+g(t) represents the combinedeffect of the thermal noise n_(th), and the sinusoidal forcing b(t) duethe dither piezo. The function f_(s), which in terms of FIG. 3 is

${f_{s} = {\frac{k}{m}\Phi}},$represents the equivalent force on the cantilever due tocantilever-sample interaction. The discretized version of the abovemodel when the sample is not present can be denoted by

$\begin{matrix}{{x_{i + 1} = {{Fx}_{i} + {Gu}_{i}}}{{y_{i} = {{Hx}_{i} + \upsilon_{i}}},{i \geq 0}}} & (4) \\{where} & \; \\{{E\left\{ {\begin{bmatrix}u_{i} \\\upsilon_{i} \\x_{i}\end{bmatrix}\begin{bmatrix}u_{j} \\\upsilon_{j} \\x_{0} \\1\end{bmatrix}}^{T} \right\}} = \begin{bmatrix}{Q_{i}\delta_{ij}} & 0 & 0 & 0 \\0 & {R_{i}\delta_{ij}} & 0 & 0 \\0 & 0 & \Pi_{0} & 0\end{bmatrix}} & (5)\end{matrix}$It is assumed that the input noise and the noise in the output areuncorrelated. Note that when the sample is not present, the cantileversettles into a sinusoidal trajectory buried in noise.

When a sample is present, there is an attractive or repulsive force onthe cantilever depending on the regime of interaction. Typically thiscantilever-sample interaction is characterized by short range repulsiveforces and long range attractive forces as shown in FIG. 4.

In many applications when the sample is introduced after the cantileverhas settled into a sinusoidal orbit, the effect of the cantilever-sampleinteraction force of block 202 is modeled as an impulsive force.Intuitively this also follows because the time spent by the cantileverunder the sample's influence is negligible compared to the time itspends outside the sample's influence. This assumption is particularlytrue for samples that have a small attractive regimes. Under such anassumption the cantilever dynamics is described byx _(i+1) =Fx _(i) +Gu _(i)+δ_(θ,i+1)νy _(i) =Hx _(i)+υ_(i) ,i≧0  (6)where θ denotes the time instant when the impact occurs, δ_(i,j) denotesthe dirac delta function, and ν signifies the magnitude of the impact.Essentially, the impact causes an instantaneous change in the state by νat time instant θ. In this setting, the time of impact and the resultingchange in the state are unknown quantities. The present inventiondetermines when the cantilever is “hitting” the sample and when it isnot, that is to detect whether the change in state is occurring or not.

In order to detect whether the change in state is occurring, an observerbased state estimation techniques and related tools are used. Note thatin an AFM setup, the position of the cantilever (deflection signal fromthe photo-diode sensors) is measurable, not the velocity. Returning toFIG. 2, cantilever 200 is the linear time invariant system depicting themodel of the cantilever. The interaction with the sample is modeled andshown as a non-linear cantilever-sample interaction system 202 appearingin the feedback path. An observer 204 is designed based on the thermalnoise and measurement noise characteristics and experimentallydetermined values of the parameters of the model as given in equation 3.

The filter 204 estimates the state of the cantilever by observing theknown input signal (e.g., sinusoidal forcing from the dither piezo) andavailable output signal (e.g., cantilever deflection data from photodiode sensor). When the sample is present as given by the model inequation 6, the estimated and filtered states from the filter 204 aregiven by,{circumflex over (x)}_(i+1|i)=F{circumflex over (x)}_(i|i),{circumflex over (x)} _(i|i) ={circumflex over (x)} _(i|i−1) +K_(i)γ_(i)  (7)with the measurement residual γ_(i), filter gain K_(i) and errorcovariance matrices given by γ_(i)=y_(i)−H{circumflex over (x)}_(i|i−1),K_(i)=P_(i|i−1)H^(T)V_(i) ⁻¹, P_(i+1|i)=FP_(i|i)F^(T)+GQG^(T),P_(i|i)=P_(i|i−1)−K_(i)HP_(i|i−1), V_(i)=HP_(i|i−1)H^(T)+R . . . In thesteady state the gain and the error covariance matrices become constantmatrices as K=lim_(i→∞)K_(i), P=lim_(i→∞)P_(i|i−1) andV=lim_(i→∞)V_(i)=HPH^(T)+R. For convenience of implementation, steadystate filter parameters are used.

The measurement residual γ_(i) is given by,γ_(i) =Y _(i;θ)ν+γ_(i) ¹  (8)where Y_(i;θ) is a known dynamic signal profile with unknown magnitude νdefined by the following recursive formulae,Y _(i;θ) =H[Φ(i,θ)−FX(i−1;θ)],X(i;θ)=K _(i) Y _(i;θ) +FX(i−1;θ),  (9)with Φ(i,θ)=Π^(i) _(j=θ)F and X(i;i)=K_(i)H. Here Y_(i;θ) and X(i;θ) arethe additive parts in the innovation and the state estimaterespectively. It can be shown that {γ_(i) ¹} is a zero mean white noisesequence with covariance V_(i) and is the measurement residual had thejump not occurred.

Note that the damping present in a typical cantilever is very low with aquality factor as high as 100 or above. This results in the cantilevertaking considerable time to settle to the steady state periodic orbit.Whenever there is an impulsive input to the cantilever (possibly due tointeraction with sample), the state changes to a new value. Since thisinput is not applied to the observer 204, the estimated state does notchange instantly. This change is fed to the observer 204 through theoutput data. The observer takes noticeable time to correct the estimatesand track the output again. This gives rise to transients in theestimation error. Note that the observer is capable of tracking thetransient response of the cantilever much before the system settles tothe steady state periodic orbit by appropriately choosing the gainK_(i). However, at the starting of the transients, there is a mismatchbetween the actual output and the estimated output. This results in thedynamic profile Y_(i;θ). This observation motivated us to develop afaster detection technique. Present detection schemes use the steadystate deflection data, whereas the approach described herein uses theresidual in innovation to probe the sample.

We pose the detection problem in the hypothesis testing framework asfollowing.H₀:Y_(i)=γ_(i) ¹, i=1,2, . . . ,nversusH ₁ :Y _(i) =Y _(i;θ)ν+γ_(i) ¹ , i=1, 2, . . . ,n  (10)where γ_(i) ¹ is a zero mean white gaussian process

${p\left( \gamma_{i}^{1} \right)} = {\frac{1}{\left( {2\pi} \right)^{\frac{n}{2}}\left( {V_{i}} \right)^{\frac{n}{2}}}{\exp\left( {{- \frac{1}{2}}\gamma_{i}^{1^{T}}V_{i}^{- 1}\gamma_{i}^{1}} \right)}}$and Y_(i;θ) is a known dynamic profile with unknown arrival time θ andunknown magnitude ν as given in equation 8. The estimation problem is tocompute the maximum likelihood estimate (MLE's) {circumflex over (θ)}(n)and {circumflex over (ν)}(n) based on the residual γ₁, . . . , γ_(n). Inone embodiment, Willsky's generalized likelihood ratio test is used tosolve both the detection and estimation problem. Willsky's generalizeratio test is described in the publications “A Generalized LikelihoodRatio Approach To Estimation In Linear Systems Subject To AbruptChanges”, Alan S. Willsky and Harold L. Jones, Proc. IEEE Decision andControl, pp. 108-112, November 1974 and “A Generalized Likelihood RatioApproach To The Detection And Estimation Of Jumps In Linear Systems”,Alan S. Willsky and Harold L. Jones, IEEE Transactions on AutomaticControl, February 1976, which are hereby incorporated by reference.

The likelihood ratio when the jump occurs at time instant θ={tilde over(θ)} of magnitude ν={tilde over (ν)} is given by

${\Lambda_{n}\left( {\overset{\sim}{\theta},\overset{\sim}{v}} \right)} = {\frac{p\left( {\gamma_{1},\ldots\mspace{14mu},\left. \gamma_{n} \middle| H_{1} \right.,{\theta = \overset{\sim}{\theta}},{v = \overset{\sim}{v}}} \right)}{P\left( {\gamma_{1},\ldots\mspace{14mu},\left. \gamma_{n} \middle| H_{0} \right.} \right)}.}$The generalized likelihood ratio computes the likelihood ratio

${\Lambda_{n} = \frac{p\left( {\gamma_{1},\ldots\mspace{14mu},\left. \gamma_{n} \middle| H_{1} \right.,{\theta = \hat{\theta}},{v = \hat{v}}} \right)}{P\left( {\gamma_{1},\ldots\mspace{14mu},\left. \gamma_{n} \middle| H_{0} \right.} \right)}},$where {circumflex over (θ)} and {circumflex over (ν)} are the maximumlikelihood estimates of θ and ν under the hypothesis H₁ (i.e.({circumflex over (θ)},{circumflex over (ν)})=argmax_(({tilde over (θ)},{tilde over (ν)}))Λ_(n)({tilde over (θ)},{tildeover (ν)})).

The decision function g_(n) defined as the double maximization of Λ_(n)over the parameters θ and ν is given by,g_(n)=max_(1≦θ≦n)d^(T)(n;θ)C⁻¹(n;θ)d(n;θ) where

${C\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i;\theta}}}$and${d\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}{Y_{i}.}}}$

The likelihood ratio is defined as l(n;θ)=d^(T)(i;θ)C⁻¹(n;θ)d(n;θ). Themaximum likelihood ratio estimate (MLE) of θ is given by the value θ≦nsuch that {circumflex over (θ)}_(n)=arg max_(1≦θ≦n)l(n;θ). The decisionrule is

$\begin{matrix}{{g_{n} = {{l\left( {n;{\hat{\theta}}_{n}} \right)}\begin{matrix}H_{1} \\ > \\ < \\H_{0}\end{matrix}ɛ}},} & (11)\end{matrix}$where the threshold ε is chosen to provide a suitable trade off betweenfalse alarm and missed alarms. The MLE of ν is given by {circumflex over(ν)}_(n)(θ)=C⁻¹(n;θ)d(n;θ). This method requires that the search forθ_(n) be on the entire data set between 1 and n, which requires a bankof filters with increasing length. In practice, the search may becarried out on a data window of finite length M. Note that the length Malso affects the missed probability. The false alarm and detectionprobabilities are calculated as,

$\begin{matrix}{P_{\; F} = {P_{\; 0}\left( \Gamma_{\; 1} \right)}} \\{= {\int_{ɛ}^{\;\infty}{{p\left( {l = {L\text{❘}H_{\; 0}}} \right)}{\mathbb{d}L}\mspace{14mu}{and}\mspace{14mu}{P_{\; D}\left( {v,\theta} \right)}}}} \\{= {P_{1}\left( \Gamma_{1} \right)}} \\{= {\int_{ɛ}^{\infty}{{p\left( {{l = {L\text{❘}H_{1}}},v,\theta} \right)}{\mathbb{d}L}}}}\end{matrix}$respectively. Since

${p\left( {l = {L\text{❘}H_{0}}} \right)} = {{- \frac{1}{2}}{\sum\limits_{i = 1}^{n}{Y_{i}^{T}V^{- 1}Y_{i}}}}$under hypothesis H₀ and Y_(i) are independent identically distributedgaussian random variables, p(l=L|H₀) is Chi-squared (χ²) density with ndegrees of freedom. Similarly

${p\left( {l = {L❘H_{1}}} \right)} = {{{- \frac{1}{2}}{\sum\limits_{i = 1}^{n}{Y_{i}^{T}V^{- 1}Y_{i}}}} - {\frac{1}{2}{v_{n}(\theta)}{\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i}}}}}$is a non-central χ² density with noncentrality parameter ν^(T)C(n;θ)ν.This shows that P_(D) is dependent upon values of θ and ν.

For specified P_(F) or P_(D), the threshold value ε can be computed fromthe tables in “Handbook of Statistical Tables”, D. B. Owen, AddisonWesley, Reading. Mass., 1922. Given ε, the values P_(F) or P_(D)(ν,θ)can be computed similarly. In one embodiment, ν is used as the minimumjump that is required to be detected and θ as the size of the datawindow M to compute P_(D).

Once a jump has been detected by a GLR detector, the MLEs {circumflexover (θ)} and {circumflex over (ν)} can be used to update the stateestimate as {circumflex over (x)}_(i|i,new)={circumflex over(x)}_(i|i,old)+{Φ(i,{circumflex over (θ)}−F└i;{circumflex over(θ)}┘}{circumflex over (ν)} and the covariance asP_(i|i,new)=P_(i|i,old)+{Φ(i,{circumflex over (θ)}−F└i;{circumflex over(θ)}┘}C⁻¹[i,{circumflex over (θ)}]{Φ(i,{circumflex over(θ)})−F└i;{circumflex over (θ)}┘}. This adaptive filtering scheme can beused to detect successive jumps. The size of the data window has to bechosen carefully to ensure the stability of the scheme.

Turning now to FIG. 5 the method of the present invention was simulatedusing Simulink in Matlab. The linear cantilever model 500 is given byequation 3, with natural frequency f₀=73,881 Hz, quality factor Q=130and a sinusoidal forcing b(t) resulting in a free oscillation amplitudeof p_(max)=24 nm. From testing, the mean deflection of the cantilever is0.3 nm due to thermal noise during normal test conditions, whichcorresponds to input noise power to the model of Q=0.001 nmHz². Theresolution of the photo-diode sensor in the set up is 1 Å, whichcorresponds to output noise power R=0.1 nm. The discrete time observer504 is designed using Matlab functions. For simplicity in thesimulation, a purely repulsive cantilever-sample interaction model 502as given in FIG. 6 is used. The cantilever interacts with the sampleonly when p(t)>1. The force from the sample is then given by f_(s)(t)=4h(t) pN. For simulation purposes, ks=4 pN/nm was chosen. This modelcorresponds to a static nonlinear block having a dead zone region oflength l. It is noted that a model that includes only the repulsive partof the cantilever-sample interaction is appropriate for many sampleswhere the attractive regime is small.

Note that the observer based state estimation is sensitive to plantmodel uncertainties and experimental conditions that depend on thermalnoise and sensor noise characteristics. In the simulations, the filter504 is designed based on the actual model 204 and noise characteristicswhile the perturbed models of the cantilever and noise characteristicsare employed as the real system.

Turning now to FIG. 7, the estimated deflection (curve 700) of thecantilever from the filter 504 is plotted with the actual deflection(curve 702). The initial estimation error seen in the first ten secondsis due to the mismatch in the initial state of the cantilever model 500and the filter 504.

In FIG. 8, the cantilever deflection signal 800 is plotted whencantilever hits the sample of 4 nm height at time instant θ=4000μsecond. It takes approximately 1200 μseconds for the cantilever toreach the steady state. In FIG. 9, the innovation sequence (curve 900)and the error in estimation (curve 902) are plotted. The dashed line incurve 900 is a 18 nm threshold level for the detection scheme of theinvention. There is an error in estimation as soon as the cantileverhits the sample. The change in the deflection is not immediately trackedby the observer 504. However, it can be seen that the observer 504 isfast enough to track the deflection signal during the transient state ofthe cantilever. It should be noted that in reality, the estimation errorapproaches a zero mean white noise much before the system stabilizes toa periodic orbit.

FIG. 10 shows a typical cantilever-sample interaction force 1000observed when the cantilever encounters a step sample profile of 4 nmhigh during simulation. This repulsive force is high in magnitude andexists for a smaller time compared to the time-period of oscillation ofthe cantilever. It is observed that the time spent by the cantileveraway from sample is 12 times of what is spent near the sample. Note thatfor analysis purposes, this force is modeled as an impulsive force (i.e.instantaneous jump in state) as previously described and is not beingdirectly applied during simulation.

In FIG. 11, the estimation error (dashed curve 1100) is plotted with thecalculated dynamic profile (solid curve 1102) when the arrival time andthe magnitude are assumed to be known. It can be seen from FIG. 11 thatthe residual innovations from the filter 504 can be modeled as a knowndynamic profile with an unknown magnitude as implied by equation 8 withadditive white Gaussian noise.

In FIG. 12 the innovation sequence is plotted when the cantilever has asingle hit with the sample of heights 2 nm (dashed curve 1200) and 4 nm(solid curve 1202). FIG. 13 plots the innovation sequence when thecantilever has multiple hits with the sample of heights 2 nm (dashedcurve 1300) and 4 nm (solid curve 1302). For the single hit case, thedynamic profiles are proportional to each other. However, this is nottrue for the multiple hits case as the dynamic profiles overlap witheach other and the arrival time of them is not uniformly separated intime.

We derived that when the cantilever hits the sample, the innovationsequence from the filter 504 consists of a dynamic profile of unknownarrival time and magnitude with additive white gaussian noise. Suchdetection and estimation problem is solved using the generalizedlikelihood ratio test (GLRT) as described above. Thus for simulation weused GLRT to detect the sample and estimate the size of impact. Theunknown arrival time and the unknown magnitude of impact is estimated inmaximum likelihood sense. The threshold ε is chosen to keep the falsealarm rate below 1%. In FIGS. 14 and 15 the generalized likelihood ratio(solid curves 1400, 1500) and the decision rule (dash-dot curves 1402,1502) are shown when the cantilever hits a repulsive sample once andmultiple times respectively with the innovation sequence (dotted curves1404, 1504). The generalized likelihood ratio plots are scaled down by afactor of 2*10⁻⁴ times to fit into the graph. The decision rule fordetection is scaled by a factor of 5 for the same reason. Note thatthere is a delay between the occurrence of the jump and its detection.The delay depends upon the threshold value ε and the size of the datawindow. From simulations, it can be seen that the occurrence isaccurately estimated after the delay.

In FIGS. 16 and 17, the estimated dynamic profiles (curves 1602, 1702)are plotted against the innovation sequence data (curves 1600, 1700)with a sample of 4 nm height. FIG. 16 shows the single hit case and FIG.17 shows the multiple hit case. Note that for FIG. 17, adaptivefiltering is used to update the innovation sequence after successivedetection and estimation. The good match between the two plots suggeststhe efficient performance of the GLRT algorithm even for the multiplehits case when the successive hits (jumps) are not widely separated intime. Note that for the multiple hit case, one has to be careful inchoosing the finite data window size to keep the false alarm and missedalarm rates low.

Conventional detection schemes rely on steady state amplitude and phasedata. In the simulation model it takes approximate 1200 μseconds toreach a steady state periodic orbit. In conventional detection schemesthat depend on demodulated amplitude profile, a threshold is chosen as0.8 times the steady state amplitude. The invention takes around 200μseconds to detect the jump considered for simulation. The transientdata exists for 60 μseconds and the decision based on GLRT is madewithin 20 μseconds. Thus, transient data based detection is at least 10times faster than the conventional case. For a single hit case, theestimation of jump is therefore 600 times faster compared to steadystate estimation.

Experiments were performed to further verify the efficacy of the newtransient signal based approach. The slowness of the piezo (the Zcomponent of the X-Y-Z scanner for sample positioning dynamics makes itdifficult to generate a waveform of pulses in the order of tens ofmicroseconds. The approach used is to make use of the piezo dynamics togenerate a testing waveform. The frequency response of the piezo wasobtained using an HP control system analyzer and a model was fit to theresponse. The model response is compared with that obtainedexperimentally in FIGS. 18 a and b. Curve 1800 is the model magnituderesponse, curve 1802 is the experimental magnitude response, curve 1804is the model phase response and curve 1804 is the experimental phaseresponse. It can be seen that the magnitude and phase responses of themodel and experimental data substantially overlap.

Simulations were performed using the model to obtain the input signalsto the piezo. FIG. 19 b shows the response of the piezo to a voltagepulse shown in FIG. 19 a of amplitude 0.5V, period 1000 μs and on time500 μs with the four peaks during the on time separated by approximately100 μs. Curve 1900 is the input voltage pulse and curve 1902 is thepiezo response. The piezo dynamics results in the occurance of 4 peaksseparated by approximately 100 μs during the on time. The maximum widthof each peak is approximately 35 μs. The cantilever is oscillated at thefirst resonant frequency of 70.1 kHz. The amplitude of oscillation wasapproximately 80 nm and the oscillating tip was approximately 100 nmaway from the sample surface. The piezo is actuated with the voltagepulse mentioned above. The oscillating cantilever interacts with theresulting peaks. The objective is to detect these peaks using thetransient signal scheme. A two mode model was obtained for thecantilever with the first resonance at 70.1 kHz and the second resonanceat 445 kHz. This model is used to build the filter and obtain theinnovation sequence. The resulting innovation sequence is shown in FIGS.20 a, 20 b and 21 a, 21 b. In FIGS. 20 a and 20 b, the cantileverdeflection signal is plotted against the pulse shape generated using thepiezo dynamics. Curve 2000 is the cantilever deflection, curve 2002 isthe piezo position, and curve 2004 is the innovations. It is difficultto detect when the cantilever tip interacts with the peaks looking atthe deflection signal 2000. The innovation process, on the other hand,bears the signature of the hits. Every time the cantilever interactswith the sample, the innovation curve loses the zero mean white natureas can be seen. In other words, the innovations become non-white whenthere is a hit and a dynamic profile appears. This dynamic profile isdetected using the techniques described above. In FIG. 21 a and b, curve2100 is the cantilever deflection, curve 2102 is the piezo position, andcurve 2104 is the innovation sequences It can be seen that theinnovations show signatures of all four peaks and that the dynamicprofile is clearly seen when the hits occur. The resulting likelihoodratio is shown in FIGS. 22 a-c. In FIGS. 22 a-c, curve 2200 is theinnovation sequence, curve 2202 is the likelihood ratio, and curve 2204is the piezo movement. Using the likelihood ratio it is possible toaccurately detect the hits as predicted by the simulations proving theefficacy of the transient signal based detection scheme.

The GLRT is a computationally expensive algorithm. A simplified versionof GLRT can be used where maximization over θ is replaced with{circumflex over (θ)}=n−M+1. Another simplification is SLGR wheremaximization over ν is replaced with a prior fixed {circumflex over(ν)}. These simplified algorithms require a wise selection of datawindow size M and prior fixed {circumflex over (ν)}.

From the foregoing, it can be seen that a new approach to determinecantilever movement has been described. An observer based stateestimation and statistical signal detection and estimation techniqueshave been applied. A first mode approximation model of the cantilever isconsidered and an observer is designed to estimate the dynamic states.The cantilever-sample interaction is modeled as an impulsive forceapplied to the cantilever in order to detect the presence of sample.Generalized likelihood ratio test is performed to obtain the decisionrule and the maximum likelihood estimation of the unknown arrival timeof the sample profile and unknown magnitude of it. Simulations on arealistic model indicate significant gains in the speed of detectionusing these techniques.

All references, including publications, patent applications, andpatents, cited herein are hereby incorporated by reference to the sameextent as if each reference were individually and specifically indicatedto be incorporated by reference and were set forth in its entiretyherein.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing the invention (especially in the context of thefollowing claims) are to be construed to cover both the singular and theplural, unless otherwise indicated herein or clearly contradicted bycontext. The terms “comprising,” “having,” “including,” and “containing”are to be construed as open-ended terms (i.e., meaning “including, butnot limited to,”) unless otherwise noted. Recitation of ranges of valuesherein are merely intended to serve as a shorthand method of referringindividually to each separate value falling within the range, unlessotherwise indicated herein, and each separate value is incorporated intothe specification as if it were individually recited herein. All methodsdescribed herein can be performed in any suitable order unless otherwiseindicated herein or otherwise clearly contradicted by context. The useof any and all examples, or exemplary language (e.g., “such as”)provided herein, is intended merely to better illuminate the inventionand does not pose a limitation on the scope of the invention unlessotherwise claimed. No language in the specification should be construedas indicating any non-claimed element as essential to the practice ofthe invention.

Preferred embodiments of this invention are described herein, includingthe best mode known to the inventors for carrying out the invention.Variations of those preferred embodiments may become apparent to thoseof ordinary skill in the art upon reading the foregoing description. Theinventors expect skilled artisans to employ such variations asappropriate, and the inventors intend for the invention to be practicedotherwise than as specifically described herein. Accordingly, thisinvention includes all modifications and equivalents of the subjectmatter recited in the claims appended hereto as permitted by applicablelaw. Moreover, any combination of the above-described elements in allpossible variations thereof is encompassed by the invention unlessotherwise indicated herein or otherwise clearly contradicted by context.

1. A method to detect transient movement of a cantilever in a sensing system, the method comprising the steps of: receiving a cantilever input signal and deflection data corresponding to the cantilever input signal; generating an estimated dynamic state of the cantilever movement using the cantilever input signal and deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude with additive white Gaussian noise; and computing maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test.
 2. The method of claim 1 wherein the step of computing maximum likelihood estimates includes the step of selecting a threshold ε such that a false alarm rate is below one percent.
 3. The method of claim 1 further comprising the step of updating the estimated dynamic state with the maximum likelihood estimates of the arrival time and the magnitude.
 4. The method of claim 1 wherein the dynamic profile is defined by the equation Y _(i;θ) =H[Φ(i,θ)−FX(i−1;θ)] X(i;θ)=K _(i) Y _(i;θ) +FX(i−1;θ) where and where Φ(i,θ)=Π^(i) _(j=θ)F, X(i;i)=K_(i)H, K_(i) is a filter gain, F is a function of the first resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever position and cantilever velocity.
 5. The method of claim 4 wherein step of generating an estimated dynamic state includes the step of generating a maximum likelihood estimate of arrival time, the maximum likelihood estimate of arrival time being given by a value θ≦n such that {circumflex over (θ)}_(n)=arg max_(1≦θ≦) _(n)l(n;θ) where l(n;θ)=d^(T)(i;θ)C⁻¹(n;θ)d(n;θ), ${{C\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i;\theta}}}},{{d\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i}}}},{V = {{HPH}^{T} + R}},$ H is a function of cantilever position and cantilever velocity, P is a state estimation error covariance matrix and R is measurement noise power.
 6. The method of claim 5 wherein the step of generating the maximum likelihood estimate of arrival time includes the step of generating the maximum likelihood estimate of arrival time in a data window of finite length.
 7. The method of claim 4 wherein the maximum likelihood estimate of magnitude is given by a value {circumflex over (ν)}_(n)(θ)=C⁻¹(n;θ)d(n;θ) where ${{C\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i;\theta}}}},{{d\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i}}}},{V = {{HPH}^{T} + R}},$ H is a function of cantilever position and cantilever velocity, P is a state estimation error covariance matrix and R is measurement noise power.
 8. The method of claim 7 wherein the step of generating the maximum likelihood estimate of magnitude includes the step of generating the maximum likelihood estimate of magnitude in a data window of finite length.
 9. A method of estimating the transient deflection of a cantilever operating in dynamic mode in a sensing system, the method comprising the steps of: receiving, by an observer, a cantilever input signal and cantilever deflection data corresponding to the cantilever input signal; generating, by the observer, an estimated dynamic state of the cantilever movement using the cantilever input signal and cantilever deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude; computing, by the observer, maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and estimating the transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude.
 10. The method of claim 9 wherein the step of generating an estimated dynamic state includes the step of generating a measurement residual, the measurement residual equal to the dynamic profile plus an offset, the offset being a non-jump measurement residual.
 11. The method of claim 10 wherein the non-jump measurement residual comprises white Gaussian noise.
 12. The method of claim 9 wherein the dynamic profile is defined by the equation Y _(i;θ) =H[Φ(i,θ)−FX(i−1;θ)] X(i;θ)=K _(i) Y _(i;θ) +FX(i−1;θ) where and where Φ(i,θ)=Π^(i) _(j=θ)F, X(i;i)=K_(i)H, K_(i) is a filter gain, F is a function of the resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever position and cantilever velocity.
 13. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value θ≦n such that {circumflex over (θ)}_(n)=arg max_(1≦θ≦n)l(n;θ) where ${{l\left( {n;\theta} \right)} = {{d^{T}\left( {i;\theta} \right)}{C^{- 1}\left( {n;\theta} \right)}{d\left( {n;\theta} \right)}}},{{C\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i;\theta}}}},{{d\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i}}}},{V = {{HPH}^{T} + R}},$ H is a function of cantilever position and cantilever velocity, P is a state estimation error covariance matrix and R is measurement noise power.
 14. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value {circumflex over (ν)}_(n)(θ)=C⁻¹(n;θ)d(n;θ) where ${{C\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i;\theta}}}},{{d\left( {n;\theta} \right)} = {\sum\limits_{i = \theta}^{n}{Y_{i;\theta}^{T}V^{- 1}Y_{i}}}},{V = {{HPH}^{T} + R}},$ H is a function of cantilever position and cantilever velocity, P is a state estimation error covariance matrix and R is measurement noise power.
 15. A control system adapted to perform the method of claim 9 comprising: a first mode approximation model of the cantilever; a cantilever-sample interaction model in communication with the first mode approximation model; and an observer designed to estimate the dynamic state of the cantilever movement, the observer in communication with the first mode approximation model of the cantilever.
 16. The control system of claim 15 wherein the cantilever-sample interaction model is modeled as an impulsive force applied to the cantilever.
 17. The control system of claim 16 wherein the impulsive force comprises a repulsive part of the cantilever-sample interaction.
 18. The control system of claim 15 wherein the first mode approximation model of the cantilever and the observer are adapted to receive an input signal of the cantilever.
 19. The control system of claim 15 wherein the observer is adapted to: receive a cantilever input signal and deflection data corresponding to the cantilever input signal; generate an estimated dynamic state of the cantilever movement using the cantilever input signal and deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude; compute maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and estimate a transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude. 